James,
If I may briefly interject, here...
While I address SR as the mechanics (mathematic mechanism) of measurement, there always exists the psychology of any individual encountering pondering what it says about reality. Much confusion arises, I think, from Eienstein's famous euphemism of riding on a beam of light and 'time stops'. uhhhh.... no. I mean if 'time' stops at light velocity, how can light continue to propagate at light velocity across space where time stops for no man? Here-in lies all the paradoxical psychologisms that plague modern relativity.
Let us dissect the Einsteinian euphemism. Starting with the old joke that the 'speed of time' is one second per second. Okay (...?). So if that is the universal rate, how do we know if it is that rate we ourselves experience? We don't. What is 'one second'? For looking closer we distinguish that for the Einsteinian euphemism to hold, it means that we track the change in rate of passage of time in relation to our velocity from the human experience of how much effort it takes to push a stone up the eternal hill, starting from rest and ever increasing our speed as we go higher up the hill. In short the euphemism rates the speed of times passage as 1 sec/sec @ rest; up to 0 sec/sec @ light velocity.
Where I find your 'A New Gamma' applicably interesting, is in looking at the Lorentzian in the reverse. That is, from light velocity dropping to ~relative rest. Mass is only a 'mass of energy' (energy en-masse) until a unit quantity specific to a unit volume is determined which exhibits inertia as a function of density in universal proportion to the total rest quantity. So gravitationally, the rate of passage of time would be 0 sec/sec at ~rest; and progress Lorentz fashion to 1 sec/sec @ light velocity. In a rest mass of particulate matter, the greatest density would exist in a core volume at constant density because time would not extend in relation to the space of that core volume, but as the distance increased from the core horizon, the density would decrease as a coeffeciency of greater space and increase of the rate of passage of time without the energy quantity extending spatially to infinity, but only to a limit minimum density where the rate of passage of time is 1 sec/sec.
Looking at it from that paradigm, SR makes better sense. At rest we measure mass, but at light velocity we measure energy. Taken literally, energy is mass existing at light velocity. And if density is a function of inherent velocity, a small enough quantity of mass does become accelerated to light velocity as EM because it ceases to be 'mass' as velocity transforms it to enegy, without the necessity theoretically to extend spatially in all directions to infinity. Hence, light is a unidirectional physical volumetric phenomenon. If I decide to produce a paper, I may well incorporate your extrapolation of (c)sqrt 1-(v^2.c^2) and cite your 'A New Gamma'. Thanks, but don't hold your breath. jrc